In an ideal case of transcription, there would be local sparsity along both the time and channel-axes because the drum events would not repeat with a rate of 1,000 Hz (which is faster than 16-beat on 240 BPM), nor would all the \(K\)  drum components be activated simultaneously. Although sparsity is one of the properties that can be achieved by the autoregressive nature of the recurrent layers, we add Sparsemax [1]} activation to encourage it explicitly. The output of Sparsemax has two important properties: i) it always sums to 1 (same as Softmax) and ii) it is highly likely to be sparse with actual zeros (unlike Softmax). In DrummerNet, two Sparsemax layers are applied in parallel, one along channel-axis (=instrument-axis) and the other time-axis within a non-overlapping window size of 64. This design choice is based on the assumption that there are only a few onsets among notes (channel-axis sparsity) and within 64 samples at \(\hat{y}\) , or 64 ms (temporal sparsity). The outputs from these two Sparsemax layers are then multiplied element-wise.
